Time series

import pandas as pd
import numpy as np

import matplotlib.pyplot as plt
import matplotlib.dates as mdates

from scipy import signal
from statsmodels.tsa.stattools import grangercausalitytests
from statsmodels.tsa.ar_model import AutoReg
from statsmodels.tsa.arima.model import ARIMA

from sklearn.metrics import r2_score, median_absolute_error, mean_absolute_error
from sklearn.metrics import median_absolute_error, mean_squared_error, mean_squared_log_error

from pathlib import Path

import os, sys
sys.path.insert(0, os.path.abspath('../src'))

from ts import *
from utils import *

data_path = "../data"

DataDownloader(data_path).download_all_csv()

covidpro_df, dpc_regioni_df, _, _, _ = load_data(data_path)

Exploration TS: infected Firenze

province = "Firenze"
compart = "New_cases"
date = 'Date'

tmp_df = covidpro_df[covidpro_df.Province == province]

visualize_ts(tmp_df, date, compart, province)

show_boxplot(tmp_df, compart)

tmp_df = tmp_df.set_index(date)

# Additive Decomposition
plot_decomposition(tmp_df, compart)

# Test for stationarity
adf_test(tmp_df[compart].values)

ADF Statistic: -3.039395882412342
p-value: 0.0313755409446347
Critical Values:
   1%, -3.45169128009473
Critical Values:
   5%, -2.8709394227049154
Critical Values:
   10%, -2.5717780602423517

# Test for trend stationarity
kpss_test(tmp_df[compart].values)

KPSS Statistic: 0.674677
p-value: 0.015848
Critical Values:
   10%, 0.347
Critical Values:
   5%, 0.463
Critical Values:
   2.5%, 0.574
Critical Values:
   1%, 0.739

# Detrend by subtracting best leastsq fit
plot_detrended_deseason(
    signal.detrend(tmp_df[compart].values), 
    province,
    compart,
    "Detrended")

# Detrend by subtracting trend comp
plot_detrended_deseason(
    remove_trend(tmp_df, compart), 
    province,
    compart,
    "Detrended")

plot_detrended_deseason_plotly(
    remove_trend(tmp_df, compart), 
    province,
    compart,
    "Detrended")

# Deseasonalize by subtracting trend
plot_detrended_deseason(
    deseason_trend(tmp_df, compart), 
    province,
    compart,
    "Deseasonalized")

# Deseasonalize by taking mov. avg. with length = seasonal window
plot_deseason_ma(tmp_df, compart, province)

# Seasonality test
plot_autocorr(tmp_df, compart)

# Autocorrelation and partial autocorrelation
plot_acf_pacf(tmp_df, compart)

"""
Lag plots:
If points get wide and scattered with increasing lag, 
this means lesser correlation
"""

plot_lag_plots(tmp_df, compart)

# Estimate forecastability
print("SampEn:", SampEn(tmp_df[compart].values, m=2, r=0.2*np.std(tmp_df[compart].values)))
print("ApproxEn:", ApEn(tmp_df[compart].values, m=2, r=0.2*np.std(tmp_df[compart].values))) 

SampEn: 0.08969363102750517
ApproxEn: 0.24777649675081914

# Smoothing
plot_smoothing(tmp_df, compart)

"""
Granger causality tests:

Infected - Month
Infected - Deaths
"""

tmp_df = covidpro_df[covidpro_df.Province == province]

tmp_df['month'] = tmp_df[date].dt.month
gc_res = grangercausalitytests(tmp_df[[compart, 'month']], maxlag=3)

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=3.0126  , p=0.0837  , df_denom=286, df_num=1
ssr based chi2 test:   chi2=3.0442  , p=0.0810  , df=1
likelihood ratio test: chi2=3.0282  , p=0.0818  , df=1
parameter F test:         F=3.0126  , p=0.0837  , df_denom=286, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=0.3696  , p=0.6914  , df_denom=283, df_num=2
ssr based chi2 test:   chi2=0.7522  , p=0.6865  , df=2
likelihood ratio test: chi2=0.7512  , p=0.6869  , df=2
parameter F test:         F=0.3696  , p=0.6914  , df_denom=283, df_num=2

Granger Causality
number of lags (no zero) 3
ssr based F test:         F=0.6856  , p=0.5615  , df_denom=280, df_num=3
ssr based chi2 test:   chi2=2.1082  , p=0.5502  , df=3
likelihood ratio test: chi2=2.1005  , p=0.5518  , df=3
parameter F test:         F=0.6856  , p=0.5615  , df_denom=280, df_num=3

gc_res = grangercausalitytests(tmp_df[[compart, 'Deaths']], maxlag=3)

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=0.7995  , p=0.3720  , df_denom=286, df_num=1
ssr based chi2 test:   chi2=0.8079  , p=0.3688  , df=1
likelihood ratio test: chi2=0.8067  , p=0.3691  , df=1
parameter F test:         F=0.7995  , p=0.3720  , df_denom=286, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=2.5788  , p=0.0776  , df_denom=283, df_num=2
ssr based chi2 test:   chi2=5.2488  , p=0.0725  , df=2
likelihood ratio test: chi2=5.2015  , p=0.0742  , df=2
parameter F test:         F=2.5788  , p=0.0776  , df_denom=283, df_num=2

Granger Causality
number of lags (no zero) 3
ssr based F test:         F=1.2330  , p=0.2980  , df_denom=280, df_num=3
ssr based chi2 test:   chi2=3.7915  , p=0.2849  , df=3
likelihood ratio test: chi2=3.7666  , p=0.2878  , df=3
parameter F test:         F=1.2330  , p=0.2980  , df_denom=280, df_num=3

Exploration TS: deaths Firenze

province = "Firenze"
compart = "Deaths"
date = 'Date'

tmp_df = covidpro_df[covidpro_df.Province == province]

visualize_ts(tmp_df, date, compart, province)

show_boxplot(tmp_df, compart)

tmp_df = tmp_df.set_index(date)

# Additive Decomposition
plot_decomposition(tmp_df, compart)

# Test for stationarity
adf_test(tmp_df[compart].values)

ADF Statistic: -2.7519413702967155
p-value: 0.06547326647232678
Critial Values:
   1%, -3.45453261164607
Critial Values:
   5%, -2.8721859575020017
Critial Values:
   10%, -2.572442854861866

# Test for trend stationarity
kpss_test(tmp_df[compart].values)

KPSS Statistic: 0.365743
p-value: 0.091921
Critial Values:
   10%, 0.347
Critial Values:
   5%, 0.463
Critial Values:
   2.5%, 0.574
Critial Values:
   1%, 0.739

# Detrend by subtracting best leastsq fit
plot_detrended_deseason(
    signal.detrend(tmp_df[compart].values), 
    province,
    compart,
    "Detrended")

# Detrend by subtracting trend comp
plot_detrended_deseason(
    remove_trend(tmp_df, compart), 
    province,
    compart,
    "Detrended")

# Deseasonalize by subtracting trend
plot_detrended_deseason(
    deseason_trend(tmp_df, compart), 
    province,
    compart,
    "Deseasonalized")

# Deseasonalize by taking mov. avg. with length = seasonal window
plot_deseason_ma(tmp_df, compart, province)

# Seasonality test
plot_autocorr(tmp_df, compart)

# Autocorrelation and partial autocorrelation
plot_acf_pacf(tmp_df, compart)

"""
Lag plots:
If points get wide and scattered with increasing lag, 
this means lesser correlation
"""

plot_lag_plots(tmp_df, compart)

# Estimate forecastability
print("SampEn:", SampEn(tmp_df[compart].values, m=2, r=0.2*np.std(tmp_df[compart].values)))
print("ApproxEn:", ApEn(tmp_df[compart].values, m=2, r=0.2*np.std(tmp_df[compart].values))) 

SampEn: 0.18732204802912877
ApproxEn: 0.47577134095104867

# Smoothing
plot_smoothing(tmp_df, compart)

"""
Granger causality tests:

Infected - Month
Infected - Deaths
"""

tmp_df = covidpro_df[covidpro_df.Province == province]

tmp_df['month'] = tmp_df[date].dt.month
gc_res = grangercausalitytests(tmp_df[[compart, 'month']], maxlag=3)

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=2.8913  , p=0.0901  , df_denom=286, df_num=1
ssr based chi2 test:   chi2=2.9216  , p=0.0874  , df=1
likelihood ratio test: chi2=2.9069  , p=0.0882  , df=1
parameter F test:         F=2.8913  , p=0.0901  , df_denom=286, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=1.6630  , p=0.1914  , df_denom=283, df_num=2
ssr based chi2 test:   chi2=3.3847  , p=0.1841  , df=2
likelihood ratio test: chi2=3.3649  , p=0.1859  , df=2
parameter F test:         F=1.6630  , p=0.1914  , df_denom=283, df_num=2

Granger Causality
number of lags (no zero) 3
ssr based F test:         F=1.0205  , p=0.3840  , df_denom=280, df_num=3
ssr based chi2 test:   chi2=3.1379  , p=0.3708  , df=3
likelihood ratio test: chi2=3.1209  , p=0.3734  , df=3
parameter F test:         F=1.0205  , p=0.3840  , df_denom=280, df_num=3

gc_res = grangercausalitytests(tmp_df[[compart, 'Deaths']], maxlag=3)

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=0.0000  , p=1.0000  , df_denom=287, df_num=1
ssr based chi2 test:   chi2=0.0000  , p=1.0000  , df=1
likelihood ratio test: chi2=-0.0000 , p=1.0000  , df=1
parameter F test:         F=485.5733, p=0.0000  , df_denom=287, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=0.0000  , p=1.0000  , df_denom=285, df_num=2
ssr based chi2 test:   chi2=0.0000  , p=1.0000  , df=2
likelihood ratio test: chi2=-0.0000 , p=1.0000  , df=2
parameter F test:         F=339.8168, p=0.0000  , df_denom=285, df_num=2

Granger Causality
number of lags (no zero) 3
ssr based F test:         F=-0.0000 , p=1.0000  , df_denom=283, df_num=3
ssr based chi2 test:   chi2=-0.0000 , p=1.0000  , df=3
likelihood ratio test: chi2=-0.0000 , p=1.0000  , df=3
parameter F test:         F=232.1738, p=0.0000  , df_denom=283, df_num=3

Exploration TS: infected Piemonte

province = "Piemonte"
compart = "nuovi_positivi"
date = 'data'

tmp_df = dpc_regioni_df[dpc_regioni_df.denominazione_regione == province]

visualize_ts(tmp_df, date, compart, province)

show_boxplot(tmp_df, compart)

tmp_df = tmp_df.set_index(date)

# Additive Decomposition
plot_decomposition(tmp_df, compart)

# Test for stationarity
adf_test(tmp_df[compart].values)

ADF Statistic: -3.3560158064650696
p-value: 0.012552855296211371
Critial Values:
   1%, -3.4540935579190495
Critial Values:
   5%, -2.8719934111688965
Critial Values:
   10%, -2.5723401594120388

# Test for trend stationarity
kpss_test(tmp_df[compart].values)

KPSS Statistic: 0.703087
p-value: 0.013265
Critial Values:
   10%, 0.347
Critial Values:
   5%, 0.463
Critial Values:
   2.5%, 0.574
Critial Values:
   1%, 0.739

# Detrend by subtracting best leastsq fit
plot_detrended_deseason(
    signal.detrend(tmp_df[compart].values), 
    province,
    compart,
    "Detrended")

# Detrend by subtracting trend comp
plot_detrended_deseason(
    remove_trend(tmp_df, compart), 
    province,
    compart,
    "Detrended")

# Deseasonalize by subtracting trend
plot_detrended_deseason(
    deseason_trend(tmp_df, compart), 
    province,
    compart,
    "Deseasonalized")

# Deseasonalize by taking mov. avg. with length = seasonal window
plot_deseason_ma(tmp_df, compart, province)

# Seasonality test
plot_autocorr(tmp_df, compart)

# Autocorrelation and partial autocorrelation
plot_acf_pacf(tmp_df, compart)

"""
Lag plots:
If points get wide and scattered with increasing lag, 
this means lesser correlation
"""

plot_lag_plots(tmp_df, compart)

# Estimate forecastability
print("SampEn:", SampEn(tmp_df[compart].values, m=2, r=0.2*np.std(tmp_df[compart].values)))
print("ApproxEn:", ApEn(tmp_df[compart].values, m=2, r=0.2*np.std(tmp_df[compart].values))) 

SampEn: 0.05778282138264353
ApproxEn: 0.22056687391795093

# Smoothing
plot_smoothing(tmp_df, compart)

"""
Granger causality tests:

Deaths - Month
Deaths - Infected
"""


tmp_df = dpc_regioni_df[dpc_regioni_df.denominazione_regione == province]

tmp_df['month'] = tmp_df[date].dt.month
gc_res = grangercausalitytests(tmp_df[[compart, 'month']], maxlag=3)

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=2.8526  , p=0.0923  , df_denom=287, df_num=1
ssr based chi2 test:   chi2=2.8824  , p=0.0896  , df=1
likelihood ratio test: chi2=2.8682  , p=0.0903  , df=1
parameter F test:         F=2.8526  , p=0.0923  , df_denom=287, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=1.0530  , p=0.3502  , df_denom=284, df_num=2
ssr based chi2 test:   chi2=2.1431  , p=0.3425  , df=2
likelihood ratio test: chi2=2.1352  , p=0.3438  , df=2
parameter F test:         F=1.0530  , p=0.3502  , df_denom=284, df_num=2

Granger Causality
number of lags (no zero) 3
ssr based F test:         F=1.2254  , p=0.3007  , df_denom=281, df_num=3
ssr based chi2 test:   chi2=3.7677  , p=0.2877  , df=3
likelihood ratio test: chi2=3.7433  , p=0.2906  , df=3
parameter F test:         F=1.2254  , p=0.3007  , df_denom=281, df_num=3

gc_res = grangercausalitytests(tmp_df[[compart, 'deceduti']], maxlag=3)

Granger Causality
number of lags (no zero) 1
ssr based F test:         F=0.0547  , p=0.8152  , df_denom=287, df_num=1
ssr based chi2 test:   chi2=0.0553  , p=0.8141  , df=1
likelihood ratio test: chi2=0.0553  , p=0.8141  , df=1
parameter F test:         F=0.0547  , p=0.8152  , df_denom=287, df_num=1

Granger Causality
number of lags (no zero) 2
ssr based F test:         F=0.0057  , p=0.9943  , df_denom=284, df_num=2
ssr based chi2 test:   chi2=0.0117  , p=0.9942  , df=2
likelihood ratio test: chi2=0.0117  , p=0.9942  , df=2
parameter F test:         F=0.0057  , p=0.9943  , df_denom=284, df_num=2

Granger Causality
number of lags (no zero) 3
ssr based F test:         F=0.0785  , p=0.9716  , df_denom=281, df_num=3
ssr based chi2 test:   chi2=0.2413  , p=0.9707  , df=3
likelihood ratio test: chi2=0.2412  , p=0.9707  , df=3
parameter F test:         F=0.0785  , p=0.9716  , df_denom=281, df_num=3

Anomalies plot

province = "Firenze"
compart = "New_cases"
date = 'Date'

tmp_df = covidpro_df[covidpro_df.Province == province]
tmp_df = tmp_df.set_index(date)

def mape(y_true, y_pred): 
    return np.mean(np.abs((y_true - y_pred) / y_true)) * 100

def adv_ma_plot(df, compart, window, plot_intervals=False, scale=1.96, plot_anomalies=False, show_anomalies_label=False, legend_position='upper left'):
    rolling_mean = df[[compart]].rolling(window=window).mean()

    plt.figure(figsize=(15,5))
    #plt.title("Moving average\n window size = {}".format(window))
    plt.plot(rolling_mean, "g", label="Rolling mean trend")

    if plot_intervals:
        mae = mean_absolute_error(df[[compart]][window:], rolling_mean[window:])
        deviation = np.std(df[[compart]][window:] - rolling_mean[window:])
        lower_bound = rolling_mean - (mae + scale * deviation)
        upper_bound = rolling_mean + (mae + scale * deviation)
        plt.plot(upper_bound, "r--", label="Upper Bound / Lower Bound")
        plt.plot(lower_bound, "r--")
        
        if plot_anomalies:
            
            anomalies = pd.DataFrame(index=df[[compart]].index, columns=df[[compart]].columns)
            anomalies[df[[compart]]<lower_bound] = df[[compart]][df[[compart]]<lower_bound]
            anomalies[df[[compart]]>upper_bound] = df[[compart]][df[[compart]]>upper_bound]
            plt.plot(anomalies, "ro", markersize=10)

            if show_anomalies_label:
                ymin, ymax = plt.ylim()
                plt.vlines(anomalies.dropna().index, ymin=ymin, ymax=ymax, linestyles='dashed', colors='grey')
                for x in anomalies.dropna().index:
                    plt.text(x, ymin + 20, x.strftime('%m-%d'), rotation=90, verticalalignment='center')
        
    plt.plot(df[[compart]][window:], label="Actual values")
    plt.legend(loc=legend_position)
    plt.show()

adv_ma_plot(tmp_df, compart, 7, plot_intervals=True, plot_anomalies=True)

adv_ma_plot(tmp_df, compart, 14, plot_intervals=True, plot_anomalies=True)

adv_ma_plot(tmp_df, compart, 30, plot_intervals=True, plot_anomalies=True)

province = "Firenze"
compart = "New_cases"
date = 'Date'

tmp_df = covidpro_df[covidpro_df.Province == province].query('20200701 > Date')
tmp_df = tmp_df.set_index(date)

adv_ma_plot(tmp_df, compart, 7, plot_intervals=True, plot_anomalies=True, show_anomalies_label=True, legend_position='upper right')

adv_ma_plot(tmp_df, compart, 14, plot_intervals=True, plot_anomalies=True, show_anomalies_label=True, legend_position='upper right')

adv_ma_plot(tmp_df, compart, 30, plot_intervals=True, plot_anomalies=True, show_anomalies_label=True, legend_position='upper right')

province = "Firenze"
compart = "New_cases"
date = 'Date'

tmp_df = covidpro_df[covidpro_df.Province == province].query('20201208 > Date > 20200701')
tmp_df = tmp_df.set_index(date)

adv_ma_plot(tmp_df, compart, 7, plot_intervals=True, plot_anomalies=True, show_anomalies_label=True, legend_position='upper left')

adv_ma_plot(tmp_df, compart, 14, plot_intervals=True, plot_anomalies=True, show_anomalies_label=True, legend_position='upper left')

adv_ma_plot(tmp_df, compart, 30, plot_intervals=True, plot_anomalies=True, show_anomalies_label=True, legend_position='upper left')

Prediction

Exponential Smoothing and Holt-Winters

province = "Firenze"
compart = "New_cases"
date = 'Date'

tmp_df = covidpro_df[covidpro_df.Province == province].query('20200701 > Date')
tmp_df.loc[(tmp_df.Date == pd.Timestamp(2020, 4, 2)), "New_cases"] = np.nan
tmp_df["New_cases"].interpolate(inplace=True)
tmp_df.loc[(tmp_df.Date == pd.Timestamp(2020, 4, 2)), "New_cases"] = np.round(tmp_df.loc[(tmp_df.Date == pd.Timestamp(2020, 4, 2)), "New_cases"])
#tmp_df[compart] = tmp_df[compart].apply(lambda x: 0 if x is np.nan or x < 0 else x)
tmp_df.astype({"New_cases": "int32"})
tmp_df = tmp_df.set_index(date)

from statsmodels.tsa.holtwinters import ExponentialSmoothing, Holt

train = tmp_df[[compart]].query('20200615 > Date')
test = tmp_df[[compart]].query('20200615 <= Date')

model = ExponentialSmoothing(train[compart].values)
model_fit = model.fit()

yhat = model_fit.predict(start=0,end=len(test)-1)

print("AIC:", model_fit.aic)
print("AICC:", model_fit.aicc)
print("BIC:", model_fit.bic)
print("k:", model_fit.k)
print("SSE:", model_fit.sse)
print("\n",model_fit.mle_retvals)

AIC: 656.2053176338434
AICC: 656.5791494095444
BIC: 661.6423153764335
k: 2
SSE: 37864.17282293216

      fun: 37864.17282293216
     jac: array([0.02880859, 0.00048828])
 message: 'Optimization terminated successfully.'
    nfev: 25
     nit: 5
    njev: 5
  status: 0
 success: True
       x: array([0.31869206, 1.05100326])

model_fit.params_formatted

	name	param	optimized
smoothing_level	alpha	0.318692	True
initial_level	l.0	1.051003	True

#plt.figure(figsize=(8,5))
#plt.plot(train.index, model_fit.fittedvalues, label='Fitted')
#plt.plot(tmp_df.index, tmp_df[compart].values, label='Real')
#plt.plot(test.index, yhat, label='Predicted')
#plt.legend()
#plt.show()


fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(train.index, model_fit.fittedvalues, label='Fitted')
ax.plot(tmp_df.index, tmp_df[compart].values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model = Holt(train[compart].values)

# It works with Powell or leastsq, but not with
# the default L-BFGS-B
model_fit = model.fit(method='Powell')

yhat = model_fit.predict(start=0,end=len(test)-1)

print("AIC:", model_fit.aic)
print("AICC:", model_fit.aicc)
print("BIC:", model_fit.bic)
print("k:", model_fit.k)
print("SSE:", model_fit.sse)
print("\n",model_fit.mle_retvals)

AIC: 657.2857815834844
AICC: 658.0857815834844
BIC: 668.1597770686648
k: 4
SSE: 36889.909956354415

    direc: array([[ 0.        ,  0.        ,  0.        ,  1.        ],
       [-0.01065353,  0.0920848 ,  0.5055263 , -0.08930595],
       [ 0.        ,  0.        ,  1.        ,  0.        ],
       [-0.00874355,  0.17655601,  0.04365726, -0.28981009]])
     fun: 36889.909956354415
 message: 'Optimization terminated successfully.'
    nfev: 238
     nit: 5
  status: 0
 success: True
       x: array([ 0.15838621,  0.22455304, -4.26962293, -0.18676495])

#plt.figure(figsize=(8,5))
#plt.plot(train.index, model_fit.fittedvalues, label='Fitted')
#plt.plot(tmp_df.index, tmp_df[compart].values, label='Real')
#plt.plot(test.index, yhat, label='Predicted')
#plt.legend()
#plt.show()

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(train.index, model_fit.fittedvalues, label='Fitted')
ax.plot(tmp_df.index, tmp_df[compart].values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_es = ExponentialSmoothing(train[compart].values)
model_fit_es = model_es.fit()

yhat_es = model_fit_es.predict(start=0,end=len(test)-1)

#plt.figure(figsize=(8,5))
#plt.plot(train.index, model_fit.fittedvalues, label='Fitted (Holt)')
#plt.plot(test.index, yhat, label='Predicted (Holt)')
#plt.plot(train.index, model_fit_es.fittedvalues, label='Fitted (ExpSmt)')
#plt.plot(test.index, yhat_es, label='Predicted (ExpSmt)')
#plt.plot(tmp_df.index, tmp_df[compart].values, label='Real')
#plt.legend()
#plt.show()

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.scatter(tmp_df.index, tmp_df[compart].values, label='Real', s=4)
ax.plot(train.index, model_fit.fittedvalues, label='Fitted (Holt)', color="orange")
ax.plot(test.index, yhat, label='Predicted (Holt)', color="orange")
ax.plot(train.index, model_fit_es.fittedvalues, label='Fitted (ExpSmt)', color="red")
ax.plot(test.index, yhat_es, label='Predicted (ExpSmt)', color="red")
plt.legend()
plt.ylabel("Number of individuals")
plt.show()

Deasonalize, detrend and make stationary

New_cases

tsplot(tmp_df[compart], lags=50)

adf_test(tmp_df[compart].values)
kpss_test(tmp_df[compart].values)

ADF Statistic: -1.3568368277900387
p-value: 0.6027853897594708
Critial Values:
   1%, -3.487517288664615
Critial Values:
   5%, -2.8865777180380032
Critial Values:
   10%, -2.5801239192052012

KPSS Statistic: 0.378572
p-value: 0.086391
Critial Values:
   10%, 0.347
Critial Values:
   5%, 0.463
Critial Values:
   2.5%, 0.574
Critial Values:
   1%, 0.739

res = tmp_df[compart] - tmp_df[compart].shift(1)
res.dropna().plot()
plt.show()

adf_test(res.dropna().values)
kpss_test(res.dropna().values)

ADF Statistic: -3.6647367715015733
p-value: 0.004637408926120581
Critial Values:
   1%, -3.487517288664615
Critial Values:
   5%, -2.8865777180380032
Critial Values:
   10%, -2.5801239192052012

KPSS Statistic: 0.158872
p-value: 0.100000
Critial Values:
   10%, 0.347
Critial Values:
   5%, 0.463
Critial Values:
   2.5%, 0.574
Critial Values:
   1%, 0.739
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/stattools.py:1876: InterpolationWarning: The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is greater than the p-value returned.

  warnings.warn(warn_msg.format(direction="greater"), InterpolationWarning)

res = np.log(tmp_df[compart]) - np.log(tmp_df[compart]).shift(1)

fig, ax = plt.subplots()
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(res.dropna())
#res.dropna().plot()
plt.show()

res_fix = res.replace([np.inf, -np.inf], np.nan).interpolate(method='time')
res_fix.plot()
plt.show()

# Shows that is stationary. Remember that KPSS null-hypo is opposite
# to ADF. We have to remove the trend and seasonality.
adf_test(res_fix.dropna().values)
kpss_test(res_fix.dropna().values)

ADF Statistic: -10.374148574924691
p-value: 2.224126550856314e-18
Critial Values:
   1%, -3.484219653271961
Critial Values:
   5%, -2.885145235641637
Critial Values:
   10%, -2.579359138917794

KPSS Statistic: 0.180195
p-value: 0.100000
Critial Values:
   10%, 0.347
Critial Values:
   5%, 0.463
Critial Values:
   2.5%, 0.574
Critial Values:
   1%, 0.739
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/stattools.py:1876: InterpolationWarning: The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is greater than the p-value returned.

  warnings.warn(warn_msg.format(direction="greater"), InterpolationWarning)

res_fix.dropna(inplace=True)

decomposed = seasonal_decompose(res_fix, model='additive', extrapolate_trend='freq')
plot_decomposition(result=decomposed)

detrended = res_fix - decomposed.trend
deseasonalized = detrended / decomposed.seasonal

fig, ax = plt.subplots()
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(deseasonalized)
plt.show()

tsplot(deseasonalized, lags=50)

adf_test(deseasonalized.values)
kpss_test(deseasonalized.values)

ADF Statistic: -10.589650606104959
p-value: 6.578190158075751e-19
Critial Values:
   1%, -3.4837793736959997
Critial Values:
   5%, -2.88495387648
Critial Values:
   10%, -2.579256976

KPSS Statistic: 0.138756
p-value: 0.100000
Critial Values:
   10%, 0.347
Critial Values:
   5%, 0.463
Critial Values:
   2.5%, 0.574
Critial Values:
   1%, 0.739
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/stattools.py:1876: InterpolationWarning: The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is greater than the p-value returned.

  warnings.warn(warn_msg.format(direction="greater"), InterpolationWarning)

AR

model = AutoReg(train, lags=1)
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(27.37116829411773, 776.8755739975904)

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(tmp_df[compart].index, tmp_df[compart].values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

model_fit.summary()

AutoReg Model Results

Dep. Variable:	New_cases	No. Observations:	112
Model:	AutoReg(1)	Log Likelihood	-526.365
Method:	Conditional MLE	S.D. of innovations	27.746
Date:	Mon, 28 Dec 2020	AIC	6.700
Time:	18:36:34	BIC	6.773
Sample:	02-25-2020	HQIC	6.730
	- 06-14-2020

	coef	std err	z	P>|z|	[0.025	0.975]
intercept	10.0673	3.423	2.941	0.003	3.359	16.776
New_cases.L1	0.6832	0.069	9.876	0.000	0.548	0.819

Roots

	Real	Imaginary	Modulus	Frequency
AR.1	1.4638	+0.0000j	1.4638	0.0000

model_fit.diagnostic_summary()

1 2.133 0.144 1 2 2.634 0.268 2 3 8.285 0.040 3 4 8.789 0.067 4 5 8.842 0.116 5 6 12.368 0.054 6 7 17.811 0.013 7 8 17.813 0.023 8 9 22.607 0.007 9 10 22.991 0.011 10 11 23.324 0.016 11

Test of No Serial Correlation

Lag	Ljung-Box	LB P-value	DF

2980.853 0.000 3.524 27.389

Test of Normality

Jarque-Bera	P-value	Skewness	Kurtosis

1 0.885 0.347 1 2 0.956 0.620 2 3 0.945 0.815 3 4 0.936 0.919 4 5 2.797 0.731 5 6 2.938 0.817 6 7 2.902 0.894 7 8 2.897 0.941 8 9 2.869 0.969 9 10 2.843 0.985 10 11 3.553 0.981 11 12 3.583 0.990 12

Test of Conditional Homoskedasticity

Lag	ARCH-LM	P-value	DF

model_fit.conf_int()

	0	1
intercept	3.358958	16.775667
New_cases.L1	0.547580	0.818741

MA

model = ARIMA(train, order=(0, 0, 1))
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(tmp_df[compart].index, tmp_df[compart].values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(29.701556472778492, 890.4492769527631)

model_fit.summary()

SARIMAX Results

Dep. Variable:	New_cases	No. Observations:	112
Model:	ARIMA(0, 0, 1)	Log Likelihood	-544.524
Date:	Mon, 28 Dec 2020	AIC	1095.049
Time:	18:49:13	BIC	1103.204
Sample:	02-24-2020	HQIC	1098.358
	- 06-14-2020
Covariance Type:	opg

	coef	std err	z	P>|z|	[0.025	0.975]
const	31.2024	6.566	4.752	0.000	18.334	44.071
ma.L1	0.5155	0.079	6.502	0.000	0.360	0.671
sigma2	975.6128	61.894	15.763	0.000	854.303	1096.922

Ljung-Box (L1) (Q):	4.87	Jarque-Bera (JB):	1793.73
Prob(Q):	0.03	Prob(JB):	0.00
Heteroskedasticity (H):	0.50	Skew:	3.31
Prob(H) (two-sided):	0.04	Kurtosis:	21.45

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

model_fit.conf_int()

	0	1
const	18.333761	44.070953
ma.L1	0.360067	0.670836
sigma2	854.303397	1096.922155

ARMA

#ARMA
model = ARIMA(train, order=(2, 0, 1))
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(tmp_df[compart].index, tmp_df[compart].values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(9.202485412359536, 89.02252350369098)

model_fit.summary()

SARIMAX Results

Dep. Variable:	New_cases	No. Observations:	112
Model:	ARIMA(2, 0, 1)	Log Likelihood	-523.022
Date:	Mon, 28 Dec 2020	AIC	1056.044
Time:	18:49:30	BIC	1069.636
Sample:	02-24-2020	HQIC	1061.559
	- 06-14-2020
Covariance Type:	opg

	coef	std err	z	P>|z|	[0.025	0.975]
const	22.2281	48.262	0.461	0.645	-72.364	116.821
ar.L1	1.1589	0.169	6.861	0.000	0.828	1.490
ar.L2	-0.1852	0.150	-1.235	0.217	-0.479	0.109
ma.L1	-0.7614	0.159	-4.804	0.000	-1.072	-0.451
sigma2	659.9514	29.979	22.014	0.000	601.193	718.709

Ljung-Box (L1) (Q):	0.06	Jarque-Bera (JB):	4387.59
Prob(Q):	0.80	Prob(JB):	0.00
Heteroskedasticity (H):	0.07	Skew:	3.83
Prob(H) (two-sided):	0.00	Kurtosis:	32.69

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

model_fit.conf_int()

	0	1
const	-72.364330	116.820508
ar.L1	0.827864	1.489956
ar.L2	-0.479207	0.108793
ma.L1	-1.072090	-0.450736
sigma2	601.193356	718.709456

ARIMA

#ARIMA
model = ARIMA(train, order=(1, 1, 1))
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(tmp_df[compart].index, tmp_df[compart].values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(3.62600149768879, 14.270879987640232)

model_fit.summary()

SARIMAX Results

Dep. Variable:	New_cases	No. Observations:	112
Model:	ARIMA(1, 1, 1)	Log Likelihood	-519.168
Date:	Mon, 28 Dec 2020	AIC	1044.335
Time:	18:53:16	BIC	1052.464
Sample:	02-24-2020	HQIC	1047.633
	- 06-14-2020
Covariance Type:	opg

	coef	std err	z	P>|z|	[0.025	0.975]
ar.L1	0.1940	0.121	1.604	0.109	-0.043	0.431
ma.L1	-0.7784	0.086	-9.000	0.000	-0.948	-0.609
sigma2	672.3067	27.742	24.234	0.000	617.933	726.680

Ljung-Box (L1) (Q):	0.04	Jarque-Bera (JB):	3552.52
Prob(Q):	0.84	Prob(JB):	0.00
Heteroskedasticity (H):	0.06	Skew:	3.34
Prob(H) (two-sided):	0.00	Kurtosis:	29.90

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

model_fit.conf_int()

	0	1
ar.L1	-0.042979	0.430930
ma.L1	-0.947893	-0.608879
sigma2	617.933265	726.680153

AR

Deseasonalized, detrended and stationary

res = np.log(tmp_df[compart]) - np.log(tmp_df[compart]).shift(1)
res_fix = res.replace([np.inf, -np.inf], np.nan).interpolate(method='time')
res_fix.dropna(inplace=True)
decomposed = seasonal_decompose(res_fix, model='additive', extrapolate_trend='freq')
detrended = res_fix - decomposed.trend
deseasonalized = detrended / decomposed.seasonal

train = deseasonalized[:'20200614']
test = deseasonalized['20200615':]

# fit model
model = AutoReg(train, lags=1)
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(1.757172058276709, 10.663912731295468)

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(deseasonalized.index, deseasonalized.values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

model_fit.summary()

AutoReg Model Results

Dep. Variable:	y	No. Observations:	126
Model:	AutoReg(1)	Log Likelihood	-413.555
Method:	Conditional MLE	S.D. of innovations	6.616
Date:	Mon, 28 Dec 2020	AIC	3.827
Time:	17:51:58	BIC	3.895
Sample:	02-27-2020	HQIC	3.855
	- 06-30-2020

	coef	std err	z	P>|z|	[0.025	0.975]
intercept	0.9854	0.599	1.646	0.100	-0.188	2.159
y.L1	0.0469	0.089	0.526	0.599	-0.128	0.222

Roots

	Real	Imaginary	Modulus	Frequency
AR.1	21.3123	+0.0000j	21.3123	0.0000

model_fit.diagnostic_summary()

1 0.621 0.431 1 2 1.275 0.529 2 3 2.527 0.470 3 4 2.731 0.604 4 5 4.710 0.452 5 6 5.795 0.447 6 7 6.320 0.503 7 8 6.398 0.603 8 9 7.473 0.588 9 10 7.534 0.674 10 11 8.698 0.650 11

Test of No Serial Correlation

Lag	Ljung-Box	LB P-value	DF

292.686 0.000 0.930 10.262

Test of Normality

Jarque-Bera	P-value	Skewness	Kurtosis

1 0.096 0.757 1 2 0.611 0.737 2 3 1.248 0.741 3 4 2.121 0.713 4 5 2.413 0.790 5 6 2.537 0.864 6 7 3.003 0.885 7 8 3.263 0.917 8 9 9.438 0.398 9 10 10.193 0.424 10 11 8.402 0.677 11 12 8.228 0.767 12

Test of Conditional Homoskedasticity

Lag	ARCH-LM	P-value	DF

model_fit.conf_int()

	0	1
intercept	-0.188076	2.158854
y.L1	-0.128060	0.221903

MA

Deseasonalized, detrended and stationary

model = ARIMA(train, order=(0, 0, 1))
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(deseasonalized.index, deseasonalized.values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(1.894738436825611, 11.226402830262469)

model_fit.summary()

SARIMAX Results

Dep. Variable:	y	No. Observations:	110
Model:	ARIMA(0, 0, 1)	Log Likelihood	-369.261
Date:	Mon, 28 Dec 2020	AIC	744.522
Time:	17:52:00	BIC	752.623
Sample:	02-26-2020	HQIC	747.808
	- 06-14-2020
Covariance Type:	opg

	coef	std err	z	P>|z|	[0.025	0.975]
const	1.2278	0.737	1.666	0.096	-0.216	2.672
ma.L1	0.0586	0.112	0.526	0.599	-0.160	0.277
sigma2	48.2271	3.326	14.502	0.000	41.709	54.745

Ljung-Box (L1) (Q):	0.00	Jarque-Bera (JB):	203.11
Prob(Q):	0.96	Prob(JB):	0.00
Heteroskedasticity (H):	1.21	Skew:	0.88
Prob(H) (two-sided):	0.57	Kurtosis:	9.42

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

model_fit.conf_int()

	0	1
const	-0.216423	2.672056
ma.L1	-0.159981	0.277204
sigma2	41.709039	54.745247

ARMA

Deseasonalized, detrended and stationary

#ARMA
model = ARIMA(train, order=(2, 0, 1))
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(deseasonalized.index, deseasonalized.values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(1.8902241050586515, 11.22901271432867)

model_fit.summary()

SARIMAX Results

Dep. Variable:	y	No. Observations:	110
Model:	ARIMA(2, 0, 1)	Log Likelihood	-368.607
Date:	Mon, 28 Dec 2020	AIC	747.213
Time:	17:53:09	BIC	760.715
Sample:	02-26-2020	HQIC	752.690
	- 06-14-2020
Covariance Type:	opg

	coef	std err	z	P>|z|	[0.025	0.975]
const	1.2285	0.698	1.759	0.079	-0.140	2.597
ar.L1	-0.4437	1.081	-0.410	0.681	-2.563	1.675
ar.L2	-0.0709	0.150	-0.474	0.635	-0.364	0.222
ma.L1	0.5034	1.122	0.449	0.654	-1.695	2.702
sigma2	47.6470	3.323	14.338	0.000	41.134	54.160

Ljung-Box (L1) (Q):	0.00	Jarque-Bera (JB):	216.18
Prob(Q):	0.97	Prob(JB):	0.00
Heteroskedasticity (H):	1.22	Skew:	0.87
Prob(H) (two-sided):	0.54	Kurtosis:	9.64

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

model_fit.conf_int()

	0	1
const	-0.140142	2.597186
ar.L1	-2.562510	1.675022
ar.L2	-0.364231	0.222360
ma.L1	-1.694703	2.701597
sigma2	41.133890	54.160118

ARIMA

Deseasonalized, detrended and stationary

#ARIMA
model = ARIMA(train, order=(1, 1, 1))
model_fit = model.fit()

# make prediction
yhat = model_fit.predict(test.index[0], test.index[-1])

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'

fig, ax = plt.subplots(figsize=(8,5))
locator = mdates.AutoDateLocator(minticks=3)
formatter = mdates.ConciseDateFormatter(locator)
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
ax.plot(model_fit.fittedvalues.index, model_fit.fittedvalues.values, label='Fitted')
ax.plot(deseasonalized.index, deseasonalized.values, label='Real')
ax.plot(test.index, yhat, label='Predicted')
plt.legend()
plt.show()

model_fit.plot_diagnostics(figsize=(10,8))
plt.plot()

[]

mean_absolute_error(test, yhat), mean_squared_error(test, yhat)

(1.8951237943879946, 11.227053295988723)

model_fit.summary()

SARIMAX Results

Dep. Variable:	y	No. Observations:	110
Model:	ARIMA(1, 1, 1)	Log Likelihood	-368.723
Date:	Mon, 28 Dec 2020	AIC	743.445
Time:	17:53:15	BIC	751.519
Sample:	02-26-2020	HQIC	746.720
	- 06-14-2020
Covariance Type:	opg

	coef	std err	z	P>|z|	[0.025	0.975]
ar.L1	0.0585	0.109	0.538	0.591	-0.155	0.272
ma.L1	-0.9992	0.589	-1.696	0.090	-2.154	0.156
sigma2	48.7290	28.428	1.714	0.087	-6.988	104.446

Ljung-Box (L1) (Q):	0.00	Jarque-Bera (JB):	161.30
Prob(Q):	0.98	Prob(JB):	0.00
Heteroskedasticity (H):	1.14	Skew:	0.78
Prob(H) (two-sided):	0.69	Kurtosis:	8.75

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

model_fit.conf_int()

	0	1
ar.L1	-0.154788	0.271799
ma.L1	-2.154180	0.155689
sigma2	-6.988208	104.446185

SARIMAX

# setting initial values and some bounds for them
from itertools import product

ps = list(range(2, 5)) + [28]
d=1 
qs = list(range(2, 5)) + [28]
Ps = range(0, 2)
D=1 
Qs = range(0, 2)
s = 7 # season length is 7

# creating list with all the possible combinations of parameters
parameters = product(ps, qs, Ps, Qs)
parameters_list = list(parameters)
len(parameters_list)

64

from tqdm import tqdm

def optimizeSARIMA(data, parameters_list, d, D, s):
    """
        Return dataframe with parameters and corresponding AIC
        
        parameters_list - list with (p, q, P, Q) tuples
        d - integration order in ARIMA model
        D - seasonal integration order 
        s - length of season
    """
    
    results = []
    best_aic = float("inf")

    for param in tqdm(parameters_list):
        # we need try-except because on some combinations model fails to converge
        try:
            model=sm.tsa.statespace.SARIMAX(data, order=(param[0], d, param[1]), 
                                            seasonal_order=(param[2], D, param[3], s)).fit(disp=-1)
        except:
            continue
        aic = model.aic
        # saving best model, AIC and parameters
        if aic < best_aic:
            best_model = model
            best_aic = aic
            best_param = param
        results.append([param, model.aic])

    result_table = pd.DataFrame(results)
    result_table.columns = ['parameters', 'aic']
    # sorting in ascending order, the lower AIC is - the better
    result_table = result_table.sort_values(by='aic', ascending=True).reset_index(drop=True)
    
    return result_table

import warnings
warnings.filterwarnings('ignore')

result_table = optimizeSARIMA(deseasonalized, parameters_list, d, D, s)

100%|██████████| 64/64 [07:31<00:00,  7.05s/it]

warnings.filterwarnings('default')

result_table.head()

	parameters	aic
0	(2, 3, 1, 1)	817.105015
1	(2, 2, 1, 1)	817.170147
2	(2, 4, 1, 1)	818.543111
3	(3, 2, 1, 1)	818.644448
4	(3, 3, 1, 1)	818.823863

#p, q, P, Q = result_table.parameters[0]
p, q, P, Q = (2, 3, 1, 1)

best_results = sm.tsa.statespace.SARIMAX(deseasonalized, order=(p, d, q), 
                                        seasonal_order=(P, D, Q, s)).fit(disp=-1)
print(best_results.summary())

/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/tsa/base/tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency D will be used.
  warnings.warn('No frequency information was'
                                      SARIMAX Results                                      
===========================================================================================
Dep. Variable:                                   y   No. Observations:                  126
Model:             SARIMAX(2, 1, 3)x(1, 1, [1], 7)   Log Likelihood                -400.553
Date:                             Fri, 11 Dec 2020   AIC                            817.105
Time:                                     16:11:14   BIC                            839.270
Sample:                                 02-26-2020   HQIC                           826.105
                                      - 06-30-2020                                         
Covariance Type:                               opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1         -1.5240      0.096    -15.919      0.000      -1.712      -1.336
ar.L2         -0.9087      0.077    -11.786      0.000      -1.060      -0.758
ma.L1          0.6194      7.419      0.083      0.933     -13.922      15.160
ma.L2         -0.6318     12.099     -0.052      0.958     -24.346      23.083
ma.L3         -0.9870      7.439     -0.133      0.894     -15.567      13.593
ar.S.L7       -0.2512      0.082     -3.073      0.002      -0.411      -0.091
ma.S.L7       -0.6741      0.079     -8.501      0.000      -0.830      -0.519
sigma2        44.1598    330.969      0.133      0.894    -604.529     692.848
===================================================================================
Ljung-Box (L1) (Q):                   0.21   Jarque-Bera (JB):                22.31
Prob(Q):                              0.65   Prob(JB):                         0.00
Heteroskedasticity (H):               0.94   Skew:                            -0.31
Prob(H) (two-sided):                  0.86   Kurtosis:                         5.04
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
/home/nick/anaconda3/lib/python3.8/site-packages/statsmodels/base/model.py:566: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

tsplot(best_results.resid, lags=50)

best_results.plot_diagnostics(figsize=(10,8))
plt.show()

best_results.conf_int()

	0	1
ar.L1	-1.711645	-1.336380
ar.L2	-1.059836	-0.757594
ma.L1	-13.921534	15.160256
ma.L2	-24.346197	23.082654
ma.L3	-15.566891	13.592808
ar.S.L7	-0.411464	-0.090980
ma.S.L7	-0.829505	-0.518677
sigma2	-604.528511	692.848032

print("MAE:", best_results.mae)
print("MSE:", best_results.mse)
print("SSE:", best_results.sse)

MAE: 4.981802952314746
MSE: 54.63265256729769
SSE: 6883.714223479509

def plotSARIMA(y_train, y_test, model, s, d):
    """
        Plots model vs predicted values
        
        series - dataset with timeseries
        model - fitted SARIMA model
        n_steps - number of steps to predict in the future
        
    """
    # adding model values
    fitted = model.fittedvalues[s+d:].copy()
    real = y_train[s+d:].copy()

    real_plus_test = pd.concat([real, y_test])
    forecast = model.predict(start=real.index[-1], end=real_plus_test.index[-1])
    fitted_plus_prediction = pd.concat([fitted, forecast])

    fig, ax = plt.subplots(figsize=(15, 7))
    locator = mdates.AutoDateLocator(minticks=3)
    formatter = mdates.ConciseDateFormatter(locator)
    ax.xaxis.set_major_locator(locator)
    ax.xaxis.set_major_formatter(formatter)
    ax.plot(fitted_plus_prediction, label="model")
    plt.axvspan(fitted.index[-1], fitted_plus_prediction.index[-1], alpha=0.5, color='lightgrey')
    ax.plot(real_plus_test, label="actual")

    #plt.figure(figsize=(15, 7))
    #plt.plot(fitted_plus_prediction, label="model")
    #plt.axvspan(fitted.index[-1], fitted_plus_prediction.index[-1], alpha=0.5, color='lightgrey')
    #plt.plot(real_plus_test, label="actual")
    plt.legend(loc='upper right')
    plt.show()

test_df = covidpro_df[covidpro_df.Province == province].query('20200715 > Date >= 20200701')
test_df[compart] = test_df[compart].apply(lambda x: 0 if x is np.nan or x < 0 else x)
test_df = test_df.set_index(date)

test_df = test_df[[compart]]

plotSARIMA(deseasonalized, test_df[[compart]][compart], best_results, s, d)

Time series as Supervised classification

tmp_df = covidpro_df[covidpro_df.Province == province].query('20200801 > Date')
tmp_df[compart] = tmp_df[compart].apply(lambda x: 0 if x is np.nan or x < 0 else x)
tmp_df = tmp_df.set_index(date)

tmp_df = tmp_df[[compart]]

data = pd.DataFrame(tmp_df[compart].copy())

data.columns = ["y"]
data["weekday"] = data.index.weekday

for i in range(1, 30):
    data["lag_{}".format(i)] = data.y.shift(i)

from sklearn.linear_model import LinearRegression
from sklearn.model_selection import cross_val_score, TimeSeriesSplit

# for time-series cross-validation set 5 folds 
tscv = TimeSeriesSplit(n_splits=5)

def timeseries_train_test_split(X, y, test_size):
    """
        Perform train-test split with respect to time series structure
    """
    
    # get the index after which test set starts
    test_index = int(len(X)*(1-test_size))
    
    X_train = X.iloc[:test_index]
    y_train = y.iloc[:test_index]
    X_test = X.iloc[test_index:]
    y_test = y.iloc[test_index:]
    
    return X_train, X_test, y_train, y_test

y = data.dropna().y
X = data.dropna().drop(['y'], axis=1)

# reserve 30% of data for testing
X_train, X_test, y_train, y_test = timeseries_train_test_split(X, y, test_size=0.3)

lr = LinearRegression()
lr.fit(X_train, y_train)

LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False)

def plotModelResults(model, X_train=X_train, X_test=X_test, plot_intervals=False, plot_anomalies=False):
    """
        Plots modelled vs fact values, prediction intervals and anomalies
    
    """
    
    prediction = model.predict(X_test)
    
    plt.figure(figsize=(15, 7))
    plt.plot(prediction, "g", label="prediction", linewidth=2.0)
    plt.plot(y_test.values, label="actual", linewidth=2.0)
    
    if plot_intervals:
        cv = cross_val_score(model, X_train, y_train, 
                                    cv=tscv, 
                                    scoring="neg_mean_absolute_error")
        mae = cv.mean() * (-1)
        deviation = cv.std()
        
        scale = 1.96
        lower = prediction - (mae + scale * deviation)
        upper = prediction + (mae + scale * deviation)
        
        plt.plot(lower, "r--", label="upper bound / lower bound", alpha=0.5)
        plt.plot(upper, "r--", alpha=0.5)
        
        if plot_anomalies:
            anomalies = np.array([np.NaN]*len(y_test))
            anomalies[y_test<lower] = y_test[y_test<lower]
            anomalies[y_test>upper] = y_test[y_test>upper]
            plt.plot(anomalies, "o", markersize=10, label = "Anomalies")
    
    error = mean_squared_error(prediction, y_test)
    plt.title("Mean squared error {0:.2f}%".format(error))
    plt.legend(loc="best")
    plt.tight_layout()
    plt.show()
    
def plotCoefficients(model):
    """
        Plots sorted coefficient values of the model
    """
    
    coefs = pd.DataFrame(model.coef_, X_train.columns)
    coefs.columns = ["coef"]
    coefs["abs"] = coefs.coef.apply(np.abs)
    coefs = coefs.sort_values(by="abs", ascending=False).drop(["abs"], axis=1)
    
    plt.figure(figsize=(15, 7))
    coefs.coef.plot(kind='bar')
    #plt.grid(True, axis='y')
    #plt.hlines(y=0, xmin=0, xmax=len(coefs), linestyles='dashed')
    plt.hlines(y=0, xmax=len(coefs))
    plt.show()

plotModelResults(lr)
plotCoefficients(lr)

from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()

y = data.dropna().y
X = data.dropna().drop(['y'], axis=1)

X_train, X_test, y_train, y_test = timeseries_train_test_split(X, y, test_size=0.3)

X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)

lr = LinearRegression()
lr.fit(X_train_scaled, y_train)

plotModelResults(lr, X_train=X_train_scaled, X_test=X_test_scaled)
plotCoefficients(lr)

from sklearn.linear_model import LassoCV, RidgeCV

ridge = RidgeCV(cv=tscv)
ridge.fit(X_train_scaled, y_train)

plotModelResults(ridge, 
                 X_train=X_train_scaled, 
                 X_test=X_test_scaled, 
                 plot_intervals=True, plot_anomalies=True)
plotCoefficients(ridge)

from xgboost import XGBRegressor 

xgb = XGBRegressor()
xgb.fit(X_train_scaled, y_train)

XGBRegressor(base_score=0.5, booster='gbtree', colsample_bylevel=1,
             colsample_bynode=1, colsample_bytree=1, gamma=0, gpu_id=-1,
             importance_type='gain', interaction_constraints='',
             learning_rate=0.300000012, max_delta_step=0, max_depth=6,
             min_child_weight=1, missing=nan, monotone_constraints='()',
             n_estimators=100, n_jobs=8, num_parallel_tree=1,
             objective='reg:squarederror', random_state=0, reg_alpha=0,
             reg_lambda=1, scale_pos_weight=1, subsample=1, tree_method='exact',
             validate_parameters=1, verbosity=None)

plotModelResults(xgb, 
                 X_train=X_train_scaled, 
                 X_test=X_test_scaled)